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Theorem equvini 1879
Description: A variable introduction law for equality. Lemma 15 of [Monk2] p. 109, however we do not require  z to be distinct from  x and  y (making the proof longer). (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.)
Assertion
Ref Expression
equvini  |-  ( x  =  y  ->  E. z
( x  =  z  /\  z  =  y ) )

Proof of Theorem equvini
StepHypRef Expression
1 equcomi 1822 . . . . . 6  |-  ( z  =  x  ->  x  =  z )
21alimi 1546 . . . . 5  |-  ( A. z  z  =  x  ->  A. z  x  =  z )
3 a9e 1817 . . . . 5  |-  E. z 
z  =  y
42, 3jctir 526 . . . 4  |-  ( A. z  z  =  x  ->  ( A. z  x  =  z  /\  E. z  z  =  y
) )
54a1d 24 . . 3  |-  ( A. z  z  =  x  ->  ( x  =  y  ->  ( A. z  x  =  z  /\  E. z  z  =  y ) ) )
6 19.29 1595 . . 3  |-  ( ( A. z  x  =  z  /\  E. z 
z  =  y )  ->  E. z ( x  =  z  /\  z  =  y ) )
75, 6syl6 31 . 2  |-  ( A. z  z  =  x  ->  ( x  =  y  ->  E. z ( x  =  z  /\  z  =  y ) ) )
8 a9e 1817 . . . . . 6  |-  E. z 
z  =  x
91eximi 1574 . . . . . 6  |-  ( E. z  z  =  x  ->  E. z  x  =  z )
108, 9ax-mp 10 . . . . 5  |-  E. z  x  =  z
1110a1ii 26 . . . 4  |-  ( A. z  z  =  y  ->  ( x  =  y  ->  E. z  x  =  z ) )
1211anc2ri 543 . . 3  |-  ( A. z  z  =  y  ->  ( x  =  y  ->  ( E. z  x  =  z  /\  A. z  z  =  y ) ) )
13 19.29r 1596 . . 3  |-  ( ( E. z  x  =  z  /\  A. z 
z  =  y )  ->  E. z ( x  =  z  /\  z  =  y ) )
1412, 13syl6 31 . 2  |-  ( A. z  z  =  y  ->  ( x  =  y  ->  E. z ( x  =  z  /\  z  =  y ) ) )
15 ioran 478 . . 3  |-  ( -.  ( A. z  z  =  x  \/  A. z  z  =  y
)  <->  ( -.  A. z  z  =  x  /\  -.  A. z  z  =  y ) )
16 nfeqf 1848 . . . 4  |-  ( ( -.  A. z  z  =  x  /\  -.  A. z  z  =  y )  ->  F/ z  x  =  y )
17 ax-8 1623 . . . . . 6  |-  ( x  =  z  ->  (
x  =  y  -> 
z  =  y ) )
1817anc2li 542 . . . . 5  |-  ( x  =  z  ->  (
x  =  y  -> 
( x  =  z  /\  z  =  y ) ) )
1918equcoms 1825 . . . 4  |-  ( z  =  x  ->  (
x  =  y  -> 
( x  =  z  /\  z  =  y ) ) )
2016, 19a4imed 1869 . . 3  |-  ( ( -.  A. z  z  =  x  /\  -.  A. z  z  =  y )  ->  ( x  =  y  ->  E. z
( x  =  z  /\  z  =  y ) ) )
2115, 20sylbi 189 . 2  |-  ( -.  ( A. z  z  =  x  \/  A. z  z  =  y
)  ->  ( x  =  y  ->  E. z
( x  =  z  /\  z  =  y ) ) )
227, 14, 21ecase3 912 1  |-  ( x  =  y  ->  E. z
( x  =  z  /\  z  =  y ) )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    \/ wo 359    /\ wa 360   A.wal 1532   E.wex 1537    = wceq 1619
This theorem is referenced by:  sbequi  1951  equvin  1999  a12lem2  28035
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1315  df-ex 1538  df-nf 1540
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